User blog:Bubby3/Extended Aarex's Graham Generator
This page shows a possible extension of Aarex's Graham Generator Nested arrays Strength: e0 The pound sign can be anything, including empty. There can be multiple pound signs #1, #2, #3 etc. In the last few pages about his notation, briefly describes dimensional arrays. Forcal12n#1(n)=Forcal1,1...1,22n-1#1(n) with n 1's. I slightly modified the definition, so in the original notation, it says n-1 instead of n 1's We can extend this to nested arrays, so 1na becomes 1n-11...1n-12na-1. We add another rule: Forcal1a#1b#2(n)=Forcal1a#12a#1b-1#2(n), and jump inside the first dimesnional seperator. 11,22= nesting 1a2, and if you are before a left-bracket, replace the a# with a#1a#...a#1a# with a a#'s, but 1a1b2 when aI(0)) The first separator is [112], which has level e0. When you see a separator with only a 1 and a level 2 or higher, search for the least separator with a level 1 less than it, It it doesn't exist, create one outside the separator with a level less than the separator, and nest it. For example, 1132 searched for level 2 brackets, or it creates a level 2 separator around it. Analysis [1122] has level psi(0)=e0 [1123] has level psi(1)=e1 [1121122] has level psi(W)=z0 [1222] has level psi(W^w)=phi(w,0) [1322] has level psi(W^W^w)=SVO [1[1[1122]22]22] has level psi(W^W^W)=LVO [1132] has level psi(e(W+1))=BHO [111,22] has level psi(Ww) We can make the notation less cumbersume by not requiring a bracket with level 2 or higher to be inside a level-1 bracket. Multi-comma arrays Strength: As strong as dropper arrays or the R function We can have ab equal to a,,b, so we can expand this 1,,1,,2 is the limit of the function, and we can go much stronger Unlike strong array notation, 1,,1,,2 can be a seperator in the main level of the array. Each seperator "searches" for a seperator, for example 1,,2 searches for _. This is almost exactly like R function. The double comma is short for 1,,,2, the triple comma is short for 1,,,,2, etc. The rules of R function and dropper arrays are too complex for me to understand, so I won't put the rules. Another rule is that if a n-uple comma is in a levels of brackets, where a1, replace the seperator, with 1a2, where a is the seperator n-a times. For example 1,,2 becomes [11,,22], and 1,,,,3 becomes [1[1[11,,,,32]2]2], when in the main array Rules: # The main rules reamin the same # If 1a1b2 becomes ab2 where b is a higher-level seperator than a, and a can be a comma # Otherwise, start the process Process: # Start at the first entry # If it is a 1, move to the right # If the entry is not a 1 ## If a sigle comma is before it, decease the entry by 1 make a copy of the whole expression with the number decreased by 1 in the entry before it ## If there are n commas before it ### Set the n commas to D, and the seperator immediatly surronding the commas to E. ### Set A to T, but decrement the array after E by 1 and replace the entry before E to a seperator sign with a 1 before it and a 2 after it. For example, A of 1,,1,,2 is 1,,1__2, and A of 1,,1,,3 is 1,,1__2,,2 ### Search for a seperator with a level less than or equal to A, and if the seperator has a lower level than A, add A inside the seperator, so 1,,1,,2 becomes [1,,11,,1,,22]. ### Set P and Q so that the seperaor inside the seperaor is PEQ ### Replace that seperator with PP...PP1QQ...QQ with the main number P's and Q's ## If there not a multple comma before it, set 1Ab to 1A2Ab-1, and jump into the first A ## If there is a left bracket before it, set 1A,#2 to 1A-1,#1A-1,#...A-1,#2 with base 1's I will do level comparison later Category:Blog posts